US7536338B2 - Method and system for automated bid advice for auctions - Google Patents
Method and system for automated bid advice for auctions Download PDFInfo
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- US7536338B2 US7536338B2 US09/955,264 US95526401A US7536338B2 US 7536338 B2 US7536338 B2 US 7536338B2 US 95526401 A US95526401 A US 95526401A US 7536338 B2 US7536338 B2 US 7536338B2
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- G—PHYSICS
- G06—COMPUTING; CALCULATING OR COUNTING
- G06Q—INFORMATION AND COMMUNICATION TECHNOLOGY [ICT] SPECIALLY ADAPTED FOR ADMINISTRATIVE, COMMERCIAL, FINANCIAL, MANAGERIAL OR SUPERVISORY PURPOSES; SYSTEMS OR METHODS SPECIALLY ADAPTED FOR ADMINISTRATIVE, COMMERCIAL, FINANCIAL, MANAGERIAL OR SUPERVISORY PURPOSES, NOT OTHERWISE PROVIDED FOR
- G06Q30/00—Commerce
- G06Q30/06—Buying, selling or leasing transactions
- G06Q30/08—Auctions
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- G—PHYSICS
- G06—COMPUTING; CALCULATING OR COUNTING
- G06Q—INFORMATION AND COMMUNICATION TECHNOLOGY [ICT] SPECIALLY ADAPTED FOR ADMINISTRATIVE, COMMERCIAL, FINANCIAL, MANAGERIAL OR SUPERVISORY PURPOSES; SYSTEMS OR METHODS SPECIALLY ADAPTED FOR ADMINISTRATIVE, COMMERCIAL, FINANCIAL, MANAGERIAL OR SUPERVISORY PURPOSES, NOT OTHERWISE PROVIDED FOR
- G06Q40/00—Finance; Insurance; Tax strategies; Processing of corporate or income taxes
- G06Q40/04—Trading; Exchange, e.g. stocks, commodities, derivatives or currency exchange
Abstract
Description
TABLE A | ||
Auction: |
Auction ID |
Auctioned Item: |
Item ID | |
Name | |
Quantity | |
Description |
Mechanism: | |
Auction Format |
Reserve price |
Bidders: |
Bidder ID | |
Segment ID |
Bids: | |
Identities of participating bidders | |
Bids submitted by participating bidders | |
Outcomes: |
Assigned payments for each bidder | ||
Assigned quantities for each bidder | ||
(Format=Dutch, Reserve Price=$100 per item, Entry Rule={By Invitation, Free}, Start Price=$500, Bid Increment=−$1/second, Closing Rule=n/a)
describes a Dutch auction in which the bidders are the ones invited to participate in the auction, the price starts at $500 and is reduced at a rate of $1 per second until a bidder claims the item at the current price. If the price reaches $100 without any bidder claiming the item, the auction ends with no sale.
(Format=Vickrey, Reserve Price=0, Entry Rule={By Invitation, Free}, Start Price=n/a, Bid Increment=n/a, Closing Rule=n/a)
describes a standard Vickrey auction with a reserve price of 0. A Vickrey auction is a sealed bid auction in which the highest bidder wins the auctioned item, and pays the second highest bid or the reserve price whichever is higher.
(Format=Sealed Bid First Price, Reserve Price=$150 per item, Start Price=n/a, Entry Rule={Open, Fee=$5}, Closing Rule=n/a)
describes a standard sealed-bid first-price auction in which anyone who pays $5 entry fee (and otherwise qualify) can submit a bid, which is required to be no less than $150, anytime during a specific time period. At the end of the specified period, the bidder who submitted the highest bid wins the item and pays what he bid. In this example all bidders are treated symmetrically.
βj (V, {Format=Vickrey, Reserve Price=R}, [2, (n 1 , F 1, (a 1 , b 1)), (n 2 , F 2, (a 2 , b 2))])=V for all bidders j.
βj (V, {Format=English, Reserve Price=R}, [2, (n 1 , F 1, (a 1 , z 1)), (n 2 , F 2, (a 2 , z 2))])=V for all bidders j.
βj (V, {Format=Sealed Bid First Price, Reserve Price=R, Bid Comparison Rule={T 1 (B), T 2 (B)}, [2, (n 1 , F 1, (a 1 , z 1)), (n 2 , F 2, (a 2 , z 2))])=φk (V) if j is in bidder group k (k=1,2).
The pair of functions (φ1 (V), φ2 (V)) is obtained by solving a system of differential equations for the unknown functions φ1 (b) and φ2 (b) and then inverting these functions to get the bid functions
φ1 (V)=φ1 −1 (V)
and
φ2 (V)=φ2 −1 (V).
The following system of differential equations in the unknown functions φ1 (b) and φ2 (b) with the initial conditions φ1 (R)=φ2 (R)=R can be solved by using any of known numerical procedures:
F 1(φ1(b))F 2(φ2(b))=[φ1(b)−b]{(n 1−1)F 2(φ2(b))F 1′(φ1(b))φ1′(b)+n2 F 1(φ1(b))F 2(φ2(b))φ2′(b)} (Eg. 1)
F 2(φ2(b))F 1(φ(b))=[φ2(b)−b]{(n 2−1)F 1(φ1(b))F 2′(φ2(b))φ2′(b)+n 1 F 2(φ2(b))F 1′(φ1(b))φ1′(b)}. (Eq. 2)
βj (V, {Format=Sealed Bid First Price, Reserve Price=R}, [2, (n 1 ,F 1,(a 1 , Z 1)), (n 2 ,F 2, (a 2 ,z 2))])=φk (V) if j is in bidder group k (k=1,2).
The pair of functions (φ1 (V), φ2 (V)) is obtained by solving a system of differential equations for the unknown functions φ1 (b) and φ2 (b) and then inverting these functions to get the bid functions φ1 (V)=φ1 −1 (V) and φ2 (V)=φ2 −1 (V). The following system of differential equations in the unknown functions φ1 (b) and φ2 (b) with the initial conditions φ1 (R)=φ2 (R)=R can be solved by using any of known numerical procedures:
[1−F 1(φ1(b))][1−F 2(φ2(b))]=[b−φ 1(b)]{(n 1−1)[1−F 2(φ2(b))]F 1′(φ1(b)) φ1′(b)+n 2[1−F 1(φ1(b))]F 2′(φ2(b))φ2′(π(b))} (Eq. 1′)
[1−F 2(φ2(b))][1−F 1(φ1(b))]=[b−φ 2(b)]{(n 2−1)[1−F 1(φ1(b))]F 2′(φ2(b)) φ2′(b)+n 1[1−F 2(φ2(b))]F 1′(φ1(b))φ1′(b)} (Eq. 2′)
-
- a set of item(s) to be auctioned (Example: A Brand New 19 inch Ultra VGA HP Monitor, A Refurbished HP Kayak Desktop Computer with
Intel Pentium III 850 Mhz, 128 M RAM, 40 G hard disk). - an evaluation criterion for ranking alternative bids (Example 1: Expected payoff. Example 2: A weighted average, or ratio, of the expected payoff to the standard deviation of the payoff).
- a set of auction the user is interested in, including a description of the item(s), a description of the rules of the auction, and a description of the number and other characteristics of rival bidders.
- initial valuations for the item(s) to be bid on (Example: bidder's valuation of {A Brand New 19 inch Ultra VGA HP Monitor, A Refurbished HP Kayak Desktop Computer with
Intel Pentium III 850 Mhz, 128 M RAM, 40 G hard disk}). - a set of relevant constraints on the bidding strategies (Example: upper and lower bounds).
Exemplary Processes
- a set of item(s) to be auctioned (Example: A Brand New 19 inch Ultra VGA HP Monitor, A Refurbished HP Kayak Desktop Computer with
βj(V, {Format=English, Reserve Price=0},[2,(5,F 1,(a 1 ,b 1)), (3,F 2,(a 2 ,b 2))])=V for all bidders j MODEL 1:
and
βj(V, {Format=Sealed Bid First Price, Reserve Price=0, [2, (5, F 1, (a 1 , b 1)), (3, F 2, (a 2 , b 2))])=φk (V) if j is in segment k (k=1,2) where (φ1(V), φ2 (V)) is the obtained from the solution to the differential equation system described above. MODEL 2:
V 1 =b+[G 1(b)G 2(b)]/[(n 1−1)G 2(b)G 1′(b)+n 2 G 1(b)G 2′(b)] (Eq. 3)
V 2 =b+[G 2(b)G 1(b)]/[(n 2−1)G 1(b)G 2′(b)+n 1 G 2(b)G 1′(b)] (Eq. 4)
where G1 (b) and G2 (b) are the distributions of bids for a bidder in segment 1 and segment 2, respectively. The equations (Eq. 3)-(Eq. 4) express the unobserved valuations in terms observable bid distributions of bidders in the two segments.
{circumflex over (β)}j (V, {Format=Sealed Bid First Price, Reserve Price=300, [2, (3, {circumflex over (F)} 1(V), (a 1 , z 1)), (4, {circumflex over (F)} 2(V), (a 2 , z 2))])=φk (V)
where φk (V) is obtained from the differential equation system described above with the initial condition φ1 (300)=φ2 (300)=300.
U(ν−transaction price)·Prob {bid b wins the auction},
or the expected payoff divided by the standard deviation of the payoff. The output may also include a table of values of the criterion at alternative (non-optimal) bids. In all the following examples, the criterion chosen is the expected payoff.
In the special case of values uniformly distributed over the interval from 0 to a, so that F(x)=x/a, and a reservation price of 0, the bids are related to values by the equation
The interpretation of this is that each bidder behaves as if he/she was risk neutral and faced (k−1)/ρ risk neutral rivals.
b=ν
that is, he/she bids as much as he/she is willing to pay. Note that this bid function does not depend on his risk attitude, the number of rivals, their values or their risk attitudes. This also applies for a Vickrey auction, where there are multiple items to be sold and the winners pay the highest losing bid.
TABLE B |
Inputs: |
Auction format: sealed bid first price | |
Reserve price: $10 | |
Number of bidders: 5 | |
Your valuation: $50 | |
Bidders' risk attitude: ρ = 0.5 | |
. . . (other characteristics of market structure/auction) |
Payoff Characteristics: |
Bid amount b | $10 | $20 | $30 | $40 | . . . |
Probability that bid b wins | .1 | .2 | .5 | .7 | . . . |
the auction | |||||
Expected payoff π | 0.63 | 1.10 | 2.24 | 2.21 | . . . |
Standard deviation of | 1.90 | 2.19 | 2.24 | 1.45 | . . . |
payoff | |||||
Maximum expected payoff π = 2.24. | |||||
Bid amount that maximizes expected payoff b = 30. |
Claims (18)
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